A general theory of purely sequential minimum risk point estimation (MRPE) of a function of the mean in a normal distribution.

A purely sequential minimum risk point estimation (MRPE) methodology with associated stopping time N is designed to come up with a useful estimation strategy. We work under an appropriately formulated weighted squared error loss (SEL) due to estimation of g (μ) , a function of μ, with g ( X ¯ N) plus linear cost of sampling from a N (μ , σ 2) population having both parameters unknown. A series of important first-order and second-order asymptotic (as c, the cost per unit sample, → 0 ) results is laid out including the first-order and second-order efficiency properties. Then, accurate sequential risk calculations are launched, which are then followed by two main results: (i) Theorem 4.1 shows an asymptotic risk efficiency property, and (ii) Theorem 5.1 shows an asymptotic second-order regret expansion associated with the proposed purely sequential MRPE strategy assuming suitable conditions on g(.). We also provide a bias-corrected version of the terminal estimator, g ( X ¯ N). We follow up with a number of interesting illustrations where Theorems 4.1–5.1 are readily exploited to conclude an asymptotic risk efficiency property and second-order regret expansion, respectively. A number of other interesting illustrations are highlighted where it is possible to verify the conclusions from Theorems 4.1–5.1 more directly with less stringent assumptions on the pilot sample size. [ABSTRACT FROM AUTHOR]